Mathematics and Its Applications

This book is designed as an introduction into what I call 'abstract' Topological Dynamics (TO): the study of topological transformation groups with respect to problems that can be traced back to the qualitative theory of differential equa­ is in the tradition of the books [GH] and [EW. The title tions. So this book (,Elements . . . ' rather than 'Introduction . . . ') does not mean that this book should be compared, either in scope or in (intended) impact, with the 'Ele­ ments' of Euclid or Bourbaki. Instead, it reflects the choice and organisation of the material in this book: elementary and basic (but sufficient to understand recent research papers in this field). There are still many challenging prob­ lems waiting for a solution, and especially among general topologists there is a growing interest in this direction. However, the technical inaccessability of many research papers makes it almost impossible for an outsider to under­ stand what is going on. To a large extent, this inaccessability is caused by the lack of a good and systematic exposition of the fundamental methods and techniques of abstract TO. This book is an attempt to fill this gap. The guiding principle for the organization of the material in this book has been the exposition of methods and techniques rather than a discussion of the leading problems and their solutions. though the latter are certainly not neglected: they are used as a motivation wherever possible.